Today, we're going to talk about Refutation by Parallel Reasoning. Now, what does that mean? In order to explain what we mean by parallel reasoning. I first have to go back and point out something that we've already learned throughout this course. What we've learned throughout this course is that, good arguments follow certain rules. There are certain rules that an argument has to follow in order to be a good argument. Some of those rules we studied when we were studying Propositional Logic. For instance you can validly infer a proposition, P, from its conjunction P & Q. You can validly infer a disjunction, P or Q, from either one of its disjuncts. You can validly infer the negation of the negation of a proposition from that proposition itself. So a lot of those rules we learned when we studied propositional logic. Other rules we learned when we studied categorical logic. For instance, if all F's are G's and no F's are G's, it follows, that there are no F's at all. If some F's are G's, and no F's are not G's, then it follows that all F's are G's. So we learn something about those rules, of valid argumentation, when we studied categorical logic. Then we learned about rules of good argumentation concerning probabilities or concerning choices. All of these different kinds of arguments follow rules and they need to follow rules in order to be good arguments but, since arguments follow rules That opens up to us, the possibility of refuting an argument, by showing that the rule that that argument follows is not a good rule of argument. In other words, if a particular argument follows a rule, we can show that that particular rule is not a good rule and thereby call into question the argument that follows that rule. Let me give you some examples to illustrate how we could do that. So, now let me give you an example of refutation by parallel reasoning. Consider the following argument. Premise 1, most solar cars have motors. In fact I think that's an understatement. I think it's true that all solar cars have motors. But even if that's true, it's at least true that most solar cars have motors. If all of them do, then most of them do. Premise two. Most cars with motors need gas. Well, that's certainly true. Most of the motorized cars in the world. At this point in time or any point in the previous time need gas and suppose you tried to persuade me on the basis of those 2 premesis. Well since most solar cars have motors and most cars with motors need gas, you conclude therefore most solar cars need gas. Now, what's wrong with that argument? Well, one way to see what's wrong with that argument is to consider the rule that that argument seems to follow. Right? For solar cars, [SOUND]. You think about solar cars as a particular category. A, call it. Then there's the category of cars that have motors. That's a second category. And to say most solar cars have motors is an example of saying most of the things that fall into one category, the As, also fall into a second category. The b's. Most a's are b's. Most solar cars have motors. But then, most of the cars with motors, most of the b's, need gas. That's to say, they fall into a third category. Things that need gas called that third category, the C's. Well, if solar cars are the category of A's and things that need gas are the category of C's. Then we can represent the conclusion this way. Most A's are C's. And so now we see that the rule that our argument follows is this. Most A's are B's. Most B's are C's. Therefore, we can conclude, most A's are C's. Now is that rule a good rule of argumentation? No, it's not. And I can prove that it's not a good rule of argumentation by considering the following example that's very obviously a bad argument. So suppose for the category of As, we use the example of, humans. Humans. Right here. And right here and suppose for the category of these, we use the example animals. Alright, cause, most humans clearly are animals. In fact, not just most humans are animals, all humans are animals. But if all humans are animals, then certainly most humans are animals. [SOUND] And for the category of C, we use the example of microscopic beings. [SOUND] See, while it's true that most of the animals that we normally think about are animals big enough for us to see Animals that are roughly our size or animals much larger, like a blue whale or animals much smaller than we are like mice or rabbits or foxes. Still the animals that we see comprise a very small portion of the animal kingdom. Most of the animal kingdom consists of animals that are too small for us to see. They're microscopic beings. So, most humans are animals. Well, that's certainly true. Most animals are microscopic. That's also true. So, on the basis of those two premises, should we then believe most humans are microscopic? Obviously not. So this is an obviously bad argument. But if this is a bad argument, then the rule that this follows, from most A's are B's and most B's are C's Draw the conclusion most A's are Cs. That rule is a bad rule of argument. In effect rule is a bad rule of argument then our original argument concerning solar cars is also a bad argument. Its's a bad argument becaus the rule that it follows. Follows is a bad rule of argument. So even if the conclusion of our original argument is true, even if it's true that most solar cars need gas, which, by the way, it isn't, but even if that were true. It would still be a bad argument for that conclusion. Remember, you can have a bad argument that has a true conclusion. And even if the conclusion of that argument were true, the argument itself would still be a bad argument because the rule that it follows is not a good rule of argument. As this example illustrates. So that's an example of refutation by parallel reasoning. We refute the original argument from most solar cars Have motors, and most motorized cars need gas, to more solar cars need gas. We refute that original argument by producing a parallel argument that's obviously unsuccessful. Even though this argument is parallel, that's to say it follows the same rule as the original argument. Now let me give you another example of refutation by parallel reasoning. Consider the following argument. It's similar to an argument that you might hear for a worldwide economic stimulus program right now. I might say, look. If I had more money, then I could buy more stuff. Therefore, if everyone had more money, everyone could buy more stuff. So I might conclude from that that governments should somehow or other, maybe through tax mitigation, maybe through rebates Somehow or other, get everyone to have more money so that everyone can buy more stuff, so that there would be more demand for businesses and so on. Okay. Now, is this a good argument? Clearly, the premise is true. If I had more money, then I could buy more stuff. But, is the argument from the premises, from the premise to the conclusion a good argument? That's the question. In order to answer that question, let's consider what the form of that argument is. You see. I could try to refute that earlier argument by parallel reasoning as follows: consider the reasoning from the premise. If I stand up let's say in a crowd of people who were all watching a certain event. Everyone in front of me is sitting down. If I stand up. Then I'll get a better view, because I'll be able to see over their heads. Now from that premise, can we draw the conclusion that if everyone stands up then everyone will get a better view? Clearly not. If everyone stands up, then no one will get a better view. You only get a better view if you stand up while everyone else is sitting down. If everyone stands up Then none of you gets a better view than you had before. So from the premise that if I stand up I will get a better view, an obviously true premise, we cannot draw the conclusion that if everyone stands up then everyone will get a better view. But if this argument is no good. Then it looks like our original argument is also no good because our original argument has the same pattern as this argument. Our original argument starts off with a premise about how things would be different for me if I had more money. And then, it draws a conclusion to the effect that things would be different in that same way for everyone if everyone had more money. But again, that doesn't follow. That doesn't follow. As this argument proves. This argument is a refutation by parallel reasoning of our original argument. Now let me show you a third example of refutation by parallel reasoning. In this third example, what we're going to be doing is not refuting the rule That an argument follows. Rather, we'll be refuting a particular premise of the argument by means of parallel reasoning. Let me show you what I mean. Consider the following reasoning. Premise 1, if God had wanted us to fly He would have given us wings. That's something I've heard people say. Premise 2, we don't have wings. So conclusion, therefore, God did not want us to fly. Now, there's nothing wrong with the rule that that argument follows. The rule that that argument follows is just Modus Ponens, which is a perfectly good argument, but there is something wrong with the first premise of that argument, as I can indicate by considering the following parallel reasoning. If God had wanted us to. Swim, he would have given us, fins. We don't have fins, obviously, and therefore, God did not want us to swim. Now, that argument seems pretty clearly unacceptable. But, maybe you don't like swimming. Maybe, more than not liking swimming, you think that swimming is an inappropriate activity for human beings. So let me change the argument yet again, and make it even more obviously unacceptable than this. How about this? If God had wanted us to stay on the ground, He would have given us roots. Well, we don't have roots. And so God did not want us to stay on the ground. Now, that Is an obvious unacceptable argument. Why is it unacceptable? It's not unacceptable because of the rule it follows, it's unacceptable because its first premise is obviously false. But if this first premise is obviously false then the first premise of our original argument if God wanted us to fly he would have given us wings. Should be equally obviously false. So that's an example of reputation by parallel reasoning in which what we refute is not the rule that the argument follows, but rather one of the premises of the argument. That concludes our lecture on Refutation by Parallel Reasoning. Which concludes our week on Refutation. Next week, we're going to be applying some of the skills we've learned. To consider various examples of arguments that arise in everyday life.